“Astronomy and mathematics are like the seasonings of wisdom.” Rabbi Eliezer ben Chisma, Pirkei Avos 4:23
When we think of the great Sages of the Talmudic (and later) eras, mathematical sophistication isn’t the first quality that comes to mind. Yet they were far ahead of their time in that regard, especially since they lived centuries before the invention of algebra. Not to make a comparison, but my mother, a”h, had the ability to solve algebraic problems using arithmetic, so we shouldn’t be surprised that the Sages could do likewise.
The first such problem that attracted my attention was a bankruptcy question that is considered in Ketubot 93a – namely, if a man has three wives, how can his estate be divided fairly if it is insufficient to fulfill all their respective ketubahs? Specifically, we’re asked to consider a case of three wives whose ketubahs call for 100 zuz, 200 zuz, and 300 zuz, respectively. Of course, if the estate equals or exceeds 600 zuz, the sum of the three obligations, there is no conflict, but what happens if it is less?
To begin, the actual statement of the Mishna reads as follows:
If a man who was married to three wives died, and the ketubah of one was a maneh, of the other two hundred zuz, and of the third three hundred zuz, and the estate [was worth] only one maneh, [the sum] is divided equally. If the estate [was worth] two hundred zuz, [the claimant] of the maneh receives fifty zuz [and the claimants respectively] of the two hundred and the three hundred zuz [receive each] three gold denarii. If the estate [was worth] three hundred zuz, [the claimant] of the maneh receives fifty zuz, and [the claimant] of the two hundred zuz [receives] a maneh, while [the claimant] of the three hundred zuz [receives] six gold denarii. Similarly, if three persons contributed to a joint fund and they had made a loss or a profit, they share in the same manner. [Note: one gold dinar = 25 zuz; 1 maneh = 100 zuz.]
The Gemara on this Mishna is unclear, referring to double seizures and other issues. Fortunately, there are contemporary sources that provide a detailed explanation of the Mishna. But let’s first consider a simpler problem: Bava Metzia 2a states: “Two hold a garment; one claims it all, the other claims half. Then the one is awarded three-fourths, the other one-fourth.” Stephen Shechter’s paper “How the Talmud Divides an Estate Among Creditors” goes on to quote: “Rashi explains the reasoning. The one who claims half concedes that half belongs to the other. Therefore, only half is in dispute. It is split equally between the two claimants.”
The Cornell blog “Game Theory in the Talmud” presents a more realistic situation. While walking down the street, Person A sees a $10 bill lying on the sidewalk. Following the rule of ye’ush (abandonment) in Chapter 2 of Bava Metzia, since the bill has no identifying marks and the person who lost it would have no way to know where it was, he has no choice but to abandon any hope of finding it. So Person A could keep the money. As he bends down to pick it up, Passerby B says he saw it too but A beat him to it, so he proposes that they split it and take $5 each. Since the contested amount is $5, A should give B half, $2.50, so he ends up with $7.50.
Shechter calls this the “Contested Garment Rule” and applies it to the following situation: “Consider an estate of size 125 with two claims on it, d1 = 100 and d2 = 200 (these were the original claims of Creditors 1 and 2). According to the Contested Garment Rule, Creditor 1 concedes 25 to Creditor 2, and Creditor 2 concedes nothing to Creditor 1. The remaining 100 is split equally between the two. Thus Creditor 1 gets 50 and Creditor 2 gets 75. These are the amounts that the Mishna awarded them.”
Before proceeding further, may I relate a personal anecdote about finding a bill on the street. My Uncle Louis, a”h, was a strictly honest man. One day as he was walking down the street in Brooklyn, a man walking ahead of him accidentally dropped a $20 bill on the sidewalk. Uncle Louis picked it up and practically ran to catch up with him and give him the money. The recipient was so grateful that he told his friends, “You see this man? Make sure nothing bad happens to him.” So that was a case where ye’ush didn’t apply.
Now we can consider the original Talmudic problem. The Talmud considers three cases (Ketubot, op.cit.) in which the estates total 100, 200, and 300 zuz, respectively, as summarized in the following table:
Our general method, now called the Talmudic Method, is to apply the Contested Garment Rule sequentially to two claimants at a time. First, when the estate is 100, all three creditors contest the entire estate, so it must be divided among them equally, 33-1/3 zuz each.
As for the third case, the contested amount between the first two claimants is 100, so each receives 50. Once the first claimant is paid, 250 is left. The contested amount between B and C is 200, so each one receives 100. Once B is paid off, 150 zuz remain for C.
The most difficult case to understand is the second, where the estate is 200 zuz. The above-referenced Cornell blog explains: “Take the 100 claimant [A] and the 200 claimant [B] with a 200 estate. Between them, they receive 125. [Why?] The 100 claimant receives 50 because of “equal division of the contested sum.” On the same estate, 150 is split equally between the 200 and 300 [C] claimants. [150 is what remains after the 100 claimant receives her 50.] They both claim a greater sum of money than 150, so the 150 is split between the two” [i.e., 75 each].
It is easy to show that the amounts paid don’t depend on the order in which the claims are processed. For example, in the third case, if we start with B and C, the contested amount is the 200 claimed by B, so each one receives 100. With B out of the picture, we’re left with 100 in the estate, and since A’s claim is 100, each receives 50. Thus, C receives 100 zuz from settling with B, and 50 zuz from settling with A, for a total of 150.
What is truly amazing is that not until 1985, more than 1,500 years later, did Israeli mathematicians Robert Aumann (who went on to win the Nobel Prize in Economics in 2005) and Michael Maschler establish the general method for solving such problems and show that the solution was unique, using game theory, a branch of mathematics developed in the 20th century. Moreover, the solution is consistent – the amount assigned is independent of what the other claimants walked away with. So, all those centuries earlier, without so much as algebra or even numerals (numbers were represented by letters of the Hebrew alphabet), Rabbi Nathan (the author of the Mishna) arrived at a correct mathematical solution.
We note that for the third case, since the estate is one-half the sum of the three claims (100 + 200 + 300 = 600), if we divide all three claims by two, we get the same solution (50, 100, 150). This is the Proportionality Rule, which can be difficult to apply, especially when there are more than three claimants. In keeping with the theme of this essay, we should mention that there are two other methods attributed to Maimonides. One is the Constrained Equal Division of Gains (Give each creditor the same amount, but don’t give any creditor more than her claim); the other is the Constrained Equal Division of Losses (Make each creditor take the same loss, but don’t make any creditor lose more than her claim). These methods are inconsistent with the Mishna.
Before leaving the topic of dividing estates or bankruptcy claims, let’s close with an example that shares little in common with the above cases other than that it involves three claimants. (As we shall see, none of the methods we’ve discussed so far can work here because there is no such animal as a fraction of a camel.) This is the story “Die Siebenzehn Kamele” (The Seventeen Camels) as recounted by Werner Maier (https://werner-maier-17-kamele.de/das-raetsel-der-17-kamele/):
It was a long time ago that a rich man lived in the Orient. He had seventeen camels; they were all his fortune. And he had three sons. When the man saw his last hour approaching, he determined: His first son gets half, his second son a third, his third son a ninth of his fortune. The only condition: No animal could be killed in the inheritance division. Then the father died.
The sons set about the division of the inheritance. And they saw: It is not possible; they could not meet the father’s condition.
When they argued with each other, there was a dervish on his camel. The sons complained of [sic] his suffering. The clever dervish considered for a moment, then said: I will help you.
He placed his camel next to the father’s seventeen camels: There were eighteen.
According to the father’s will, he gave half to the first son, nine camels; the second son a third, six camels; the third son a ninth, that was two camels.
9 + 6 + 2 makes 17 camels. There is 1 camel left: It was his.
The dervish took it and happily went his way. The sons were happy.
Aren’t mathematical puzzles fun?
